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By Huang X., Yin W.
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Extra resources for A Bishop surface with a vanishing Bishop invariant
Duke Math. J. 32, 1–21 (1965) 5. : Sur les variétés pseudo-conformal des hypersurfaces de l’espace de deux variables complexes. Ann. Mat. Pura Appl. 11(4), 17–90 (1932) 6. : Real hypersurfaces in complex manifolds. Acta Math. 133, 219–271 (1974) 7. : Filling by holomorphic discs and its applications. In: Geometry of Low-Dimensional Manifolds. Lond. Math. Soc. Lect. , vol. 151 (1997) 8. : Most real analytic Cauchy–Riemann manifolds are nonalgebraizable. Manuscr. Math. 115, 489–494 (2004) 9. : On the convergence of normalizations of real analytic surfaces near hyperbolic complex tangents.
54) Here, we used the obvious fact that the Euclidean length of ∂S(u) is bounded by a constant. 53). Assume that z is on the hyperbolic geodesic segment in D(u) connecting Aj (u) to Aj+1 (u) for a certain j ∈ [0, s − 1]. ) X. Huang, W. 55) j+1 for a certain t ∈ [0, 1]. 22), we get |z(u, t)| = s · (s − 1) 1−s s |1 + t(e √ 2π −1 s − 1)|u s−1 s + o(u s−1 s ). 56) for 0 < u 1. 14. 11 as follows. We notice that √ √ (i) σ ∗ (ζ, u) has a convergent power series expansion in (ζ, u) near (0, 0),√ 1 (ii) Ψ(τ, u)√has a convergent power series expansion in τ and u 2s and, √ z −1 √z (iii) σ ( u , u) has a convergent power series expansion in ( √u , u), too.
Since there are polynomials G 1 and G 2 such that D X (P(z, z, X )) X=H(N ) ∗ N G 1 P + G 2 PX = R and since P(z, z, H(N ) ) = o(|z| ), we conclude that ∗ the degree k0 of the lowest non-vanishing order term of PX (z, z, H(N ) ) is bounded by C1 (d), depending only on d. Choose an N > C1 (d) and a sufficiently small positive number δ. 23) to conclude that each aα0 β0 with α0 +β0 ≥ N is determined by bαβγ and aαβ with α+β ≤ N −1 through rational functions in aαβ (α + β ≤ N − 1) and bαβγ (α + β + γ ≤ d) with at most C(k0 , d, N ) variables, here C(k0 , d, N ) depends only on k0 , d, N.
A Bishop surface with a vanishing Bishop invariant by Huang X., Yin W.